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## Fluid Mechanics

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States of Matter

- Solids: definite shape, definite volume
- Liquids: indefinite shape, definite volume
- Gases: indefinite shape, indefinite volume
- Liquids & gases are collectively called “fluids”
- Fluid: nonsolid state of matter in which the atoms or molecules are free to move past each other

Density

- In the study of fluid mechanics mass density is an important quantity.
- concentration of matter of an object
- measured as the mass per unit volume of a substance
- represented by Greek letter (rho)
- units: kg/m3
- Other forms of density: linear, energy
- Solids & liquids are almost incompressible, so the mass density is independent of pressure.
- Gases are compressible and their density changes based on temperature and pressure.

Buoyancy

- Why do some objects float and other objects sink?
- Why can a submarine and ocean creatures remain at a certain depth below the water , then sink or rise?
- Why do things feel lighter underwater than they do in air?

Buoyant Force

- When an object is immersed in or is floating on a liquid, there is a buoyant force exerted upward on it.
- The net force acting on a floating object is zero, so the buoyant force equals the object’s weight.
- The net force acting on a submerged object is the difference between the object’s weight and the buoyant force.
- This net force is called the apparent weight

Archimedes’ Principle

- Archimedes was an ancient Greek mathematician from the city of Syracuse on the island of Sicily. The king of Syracuse was concerned that his supposedly gold crown was a fake. Archimedes figured out how to test the crown’s authenticity while bathing, thereby discovering the principle of buoyancy, now known as Archimedes’ principle. He was reported to have been so excited by this discovery that he immediately jumped out of the bath and ran through the streets of Syracuse, sans bathrobe, shouting “Eureka!”, meaning “I’ve found it!”
- Archimedes’ Principle:any object completely or partially submerged in a fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the object

Archimedes’ Principle

- When an object is submerged in a liquid, it displaces a volume of the liquid equal to its own volume.
- If it is only partially submerged, it displaces a volume of the liquid equal to the volume of the object that is submerged.
- Regardless, the weight of the displaced fluid is equal to the buoyant force exerted upward on the submerged object.

Floating Objects

- For a floating object, the net force on the object is zero.
- Therefore the buoyant force must equal the weight of the object.

Apparent Weight

- Apparent weight is the vector sum of the buoyant force and the weight of the object.
- Ultimately depends on the relative density of the object and the fluid.

Fluid Pressure

- Pressure: magnitude of the force on a surface per unit area
- SI Unit: pascal (Pa)
- 1 Pa = 1 N/m2
- ~105 Pa = 1 atm

Pascal’s Principle

- Blaise Pascal (1623-1662, French)
- Pascal’s Principle:
- pressure applied to a fluid in a closed container is transmitted equally to every point of the fluid and to the walls of the container
- if pressure is increased at any point in a container, the pressure increases at all point inside the container by exactly the same amount
- Examples/Applications:
- Bicycle pump
- Hydraulic lifts, pistons, and other hydraulic machines

Pressure Varies With Depth

- Fluid at a given depth must support the weight of fluid above it
- As depth increases, fluid must support a greater weight of fluid above it
- Therefore, pressure increases with depth

Pressure Varies With Depth

Volume of water above submarine:

A

Mass of column of water:

h

Weight of column of water:

Pressure at this depth:

Imagine this is a small area of a submarine.

hg is called the gauge pressure

Note: This equation is only valid if the density ofthe fluid is uniform over the distance, h.

Fluid Pressure as a Function of Depth

- Gauge pressure, hg, is not the total pressure at that depth.
- The atmosphere also exerts a pressure, P0, at the surface that is transmitted to that depth.
- The total, or absolute pressure, P, is equal to the sum of the gauge pressure and the atmospheric pressure.

Buoyancy and Fluid Pressure

- Buoyant forces arise from differences in fluid pressure between the top and bottom of an immersed object.

Fluid Flow

- When a fluid is in motion, the flow can be characterized as either:
- Laminar flow
- smooth flow
- all particles follow the same smooth path
- predictable, easy to model
- Turbulent flow
- irregular flow
- abrupt changes in velocity
- occurs when the fluid flow exceeds a certain velocity
- can also be caused by obstacles and sharp turns
- irregular motions called eddy currents
- extremely chaotic, unpredictable

Ideal Fluids

- When modeling fluid motion, we consider ideal fluids.
- Nonviscous (no internal friction)
- Incompressible (uniform density)
- Characterized by steady flow (constant velocity and pressure)
- Nonturbulent
- In reality, no fluid is ideal, but ideal fluids can help explain properties of real fluids.

Continuity Equation

- An ideal fluid flows into one end of a pipe and out the other end.
- If the diameter of the pipe is different at each end, how does the speed of fluid flow change as the fluid passes through the pipe?
- Mass must be conserved, so…

Flow Rate

- The product, Av, is called the flow rate and has units of volume per unit time.
- The flow rate is constant throughout the pipe.
- If the cross-sectional area is small (narrow pipe), the velocity will be great.
- If the cross-sectional area is large (wide pipe), the velocity will be low.
- Examples:
- Stream of water from a faucet narrows as it falls (accelerates due to gravity)
- Rivers flow more rapidly in places where the river is shallow or narrow

Bernouli’s Principle

- If the velocity of a fluid increases, as the pipe narrows, there must be a net force causing the acceleration.
- This unbalanced force is due to a difference in water pressure in front and behind. The pressure behind must be greater than the pressure in front.

- Bernouli’s Principle:
- the pressure in fluid decreases as the fluid’s velocity increases
- Lift on an airplane wing
- Air flows faster over the wing than under it
- Pressure above the wing is less than under it
- Net force upward causes lift

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